Geoscience Reference
In-Depth Information
What happens when statistical techniques that assume independence of
observations are used in situations that there is spatial autocorrelation? In general,
the results will be less precise than those obtained by geostatistical methods,
variances will be significantly underestimated, and 95 % confidence intervals will
be too narrow. Whether or not these shortcomings are serious depends on the
characteristics of the subject of application. It is possible that simple averaging
provides good results especially in situations where there is the possibility of
unforeseen events that cannot be observed immediately. Experienced mining
geologists and engineers will know that results obtained by means of exploratory
boreholes or channel sampling have limited value depending on circumstances.
Geostatistics in mining is most valuable if it is possible and necessary to process
large amounts of data that have similar properties such as the hundreds of thousands
of gold and uranium determinations in the Witwatersrand goldfields in South
Africa. Although the theory of geostatistics primarily was commenced in the field
of mining, it should be appreciated that to-day there are many other applications of
Matheron's original approach. There now exist numerous applications in environ-
mental sciences, agriculture, meteorology, oceanography, physical geography
and other fields. Geostatistical textbooks include Deutsch (
2002
), Goovaerts
(
1997
) and Olea (
1999
). Recent new developments in the theory of geostatistics
include multiple-point geostatistical simulation based on genetic algorithms
(Peredo and Ortiz
2012
), sequential simulation with iterative methods (Arroyo
et al.
2012
) and extensions of the parametric inference of spatial covariances by
maximum likelihood (Dowd and Pardo-Ig
´
zquiza
2012
).
Box 2.1: Basic Elements of Classical Statistics
ðÞ
¼
R
1
1
0
r
¼
EX
r
x
r
fx
The
r
-th moment of a random variable
X
is
μ
ðÞ
dx
where
E
denotes mathematical
expectation. The mean
of
X
satisfies
0
R
1
1
0
1
¼
μ
¼
μ
EX
ðÞ
¼
xf x
ðÞ
dx
. Moments about
the mean are defined as:
E
X
μ
r
0
R
1
1
r
fx
μ
r
¼
ð
x
μ
Þ
ðÞ
dx
.If
c
is a constant,
E
(
X
+
c
)
EX
+
c
¼
¼
and
E
(
cX
)
cEX
. When
X
and
Y
are two random variables with
two-dimensional frequenc
y
distribution
f
(
x
,
y
),
E
(
X
+
Y
)
¼
¼
R
1
1
R
1
1
(
x
+
y
)
EX
+
EY
.If
X
is a sample mean,
E X
f
(
x
,
y
)
dxdy
¼
¼
μ
.If
X
is a binary
variable with probabilities
P
(
X
¼1) ¼
a
and P(
X
¼0) ¼1-
a
where
a
is
a constant,
then
E
(
X
)
¼
P
(
X
¼
1)
¼
a
. Consequently, probabilities can
be
treated
as
expected
values.
The
variance
of
X
is
E
X
μ
2
¼
R
1
1
2
fx
2
X
0
2
2
. Its properties
˃
ðÞ
¼
μ
2
¼
ð
x
μ
Þ
ðÞ
dx
¼
μ
μ
2
(
X
+
c
)
2
(
X
) and
2
(
cX
)
c
2
2
(
X
). If
X
and
Y
are independent, E
include
˃
¼
˃
˃
¼
˃
2
(
X
+
Y
)
2
(
X
)+
2
(
Y
). If
X
is t
he
mean of a sample
(
XY
)
¼
(
EX
)(
E
Y
) and
˃
¼
˃
˃
2
X
¼
˃
¼
2
X
2
X
n
2
X
of size
n
,
. The
factor
n
/(
n
-1) is known as Bessel's correction. Switching to conventional
˃
ðÞ=
n
. It also follows that
˃
X
1
˃
ðÞ
n
X
n
i
¼1
x
i
x
2
ð
Þ
sample notation, it follows that,
s
2
ðÞ
¼
x
.
n
1
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